Hartree-Fock calculations, momentum density

Hartree-Fock calculations of the three leading coefficients in the MacLaurin expansion, Eq. (5.40), have been made [187,232] for all atoms in the periodic table. The calculations [187] showed that 93% of rio(O) comes from the outermost s orbital, and that IIo(O) behaves as a measure of atomic size. Similarly, 95% of IIq(O) comes from the outermost s and p orbitals. The sign of IIq(O) depends on the relative number of electrons in the outermost s and p orbitals, which make negative and positive contributions, respectively. Clearly, the coefficients of the MacLaurin expansion are excellent probes of the valence orbitals. The curvature riQ(O) is a surprisingly powerful predictor of the global behavior of IIo(p). A positive IIq(O) indicates a type 11 momentum density, whereas a negative rio(O) indicates that IIo(O) is of either type 1 or 111 [187,230]. MacDougall has speculated on the connection between IIq(O) and superconductivity [233]. 

Duncanson and Coulson [242,243] carried out early work on atoms. Since then, the momentum densities of aU the atoms in the periodic table have been studied within the framework of the Hartree-Fock model, and for some smaller atoms with electron-correlated wavefunctions. There have been several tabulations of Jo q), and asymptotic expansion coefficients for atoms [187,244—251] with Hartree-Fock-Roothaan wavefunctions. These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253]. There have also been several reports of electron-correlated calculations of momentum densities, Compton profiles, and momentum moments for He [236,240,254-257], Li [197,237,240,258], Be [238,240,258, 259], B through F [240,258,260], Ne [239,240,258,261], and Na through Ar [258]. Schmider et al. [262] studied the spin momentum density in the lithium atom. A review of Mendelsohn and Smith [12] remains a good source of information on comparison of the Compton profiles of the rare-gas atoms with experiment, and on relativistic effects. 

Values of the MacLaurin coefficients computed from good, self-consistent-field wavefunctions have been reported [355] for 125 linear molecules and molecular ions. Only type I and II momentum densities were found for these molecules, and they corresponded to negative and positive values of IIq(O), respectively. An analysis in terms of molecular orbital contributions was made, and periodic trends were examined [355]. The qualitative results of that work [355] are correct but recent, purely numerical, Hartree-Fock calculations [356] for 78 diatomic molecules have demonstrated that the highly regarded wavefunctions of Cade, Huo, and Wahl [357-359] are not accurate for IIo(O) and especially IIo(O). These problems can be traced to a lack of sufficiently diffuse functions in their large basis sets of Slater-type functions. 

In summary, structure calculations can obtain 1 or 2% agreement with accurate optical data. A broader perspective is given in chapter 11 by electron momentum spectroscopy. Hartree—Fock calculations agree with one-electron momentum densities within experimental error, but configuration-interaction calculations agree only qualitatively with detailed data on correlations. 

Numerical Hartree-Fock calculations, free from basis set artifacts, have been used to establish that the ground state momentum densities of all the atoms and their ions can be classihed into three types [84,85]. Type I and III momentum densities are found almost exclusively in metal atoms He, N, all atoms from groups 1-14 except Ge and Pd, and all the lanthanides and actinides. These momentum densities all have a global maximum at p = 0 and resemble the momentum density shown in Fig. 19.3 for the beryllium atom. The maximum atp = 0 comes mainly from the outermost s-subshell, 2s in this case. Type I and III densities dilfer in that the latter have a secondary maximum that is so small as to be invisible on a diagram such as Fig. 19.3. Type II densities are the norm for non-metallic atoms and are found in Ge, Pd and all atoms from groups 15-18 except He and... 

Secondly, correlations in the initial state can lead to experimental orbital momentum densities significantly different from the calculated Hartree-Fock ones. Figure 3 shows such a case for the outermost orbital of water, showing how electron-electron correlations enhance the density at low momentum. Since low momentum components correspond in the main to large r components in coordinate space, the importance of correlations to the chemically interesting long range part of the wave function is evident. 

Another manifestation of the reciprocity of densities in r- and p-space is provided by Fig. 19.2. It shows the radial electron number density D r) = Aiir pir) and radial momentum density /(p) = Aitp nip) for the ground state of the beryllium atom calculated within the Hartree-Fock model in which the Be ground state has a ls 2s configuration. Both densities show a peak arising from the Is core electrons and another from the 2s valence electrons. However, the origin of the peaks is reversed. The sharp,... 

Finally, we consider density functional theory (DFT) computations of p-space properties. A naive way of calculating p-space properties is to use the Kohn-Sham orbitals obtained from a DFT computation to form a one-electron, r-space density matrix Fourier transform / according to Eq. (14), and proceed further. This approach is incorrect because the Kohn-Sham density matrix F is not the true one and, in fact, corresponds to a fictitious non-interacting system with the same p(r) as the true system. On the other hand, Hamel and coworkers [112] have shown that if the exact Kohn-Sham exchange potential is used, then the spherically averaged momentum densities of the Kohn-Sham orbitals should be very close to those of the Hartree-Fock orbitals. Of course, in practical computations the exact Kohn-Sham exchange potential is not used since it is generally not known. 

In practice, using currently available exchange and correlation potentials, this path leads to results [113] worse than those obtained with the Hartree-Fock method. This is illustrated for momentum moments in Table 19.2 which shows median absolute percent errors of (p ) for 78 molecules relative to those computed by an approximate singles and doubles coupled-cluster method often called QCISD [114,115]. The molecules are mostly polyatomic, and contain H, C, N, O, and F atoms. The correlation-consistent cc-pVTZ basis set [110] was used for these computations. Table 19.2 shows the median errors for the Hartree-Fock method, for second-order Mpller-Plesset permrbation theory (MP2), and for DFT calculations done with the B3LYP hybrid density functional... 

Other calculations tested using this molecule include two-dimensional, fully numerical solutions of the molecular Dirac equation and LCAO Hartree-Fock-Slater wave functions [6,7] local density approximations to the moment of momentum with Hartree-Fock-Roothaan wave functions [8] and the effect on bond formation in momentum space [9]. Also available are the effects of information theory basis set quality on LCAO-SCF-MO calculations [10,11] density function theory applied to Hartree-Fock wave functions [11] higher-order energies in... 

The [BHJ- ion has been used as a model to probe electron densities and the work indicates that the Hartree-Fock method can be used to investigate the corresponding proton densities in coordinate space and in momentum space [10]. The NMR chemical shift and shift derivative based on bond extension have been calculated for the [BHJ- ion to be 154.1 ppm and -27 ppm/A, respectively, using the GIAO SCF approach [11]. 

The quantity (f) q) represents a two-body density obtained for a given approximation to calculate < F>. For this two-body density all variables have been integrated except the momentum transfer. Therefore this 4>(q) is a measure to which extent the local interaction at momentum transfer q contributes to the total interaction energy. In the BHF approximation this (q) can be split into a direct part (related to the Hartree contribution) and an exchange contribution (related to the Fock term)... 

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